ferguson - variability

8/8/2019 Ferguson - Variability

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OF VARIATION,SKEWNESS.

MEASURES

AND KURTOSIS

5.I INTRODUCTIONOf greatcOncernio the statistician is the var-iation in the events of nature.

The variatiDn of one measuremenl from another is a persislingcharacter

istic of any sample of measurements. Measurements of intelligence, eye

color, reaction time, and skin resistance, for example, exhibit variation in

any sample of individuals. Anthropometric measurementssuch as height.

weight,diameterof the skull. length of the forearm, and angularseparation

of the metatarsals show variation betwe'enindividuals. Anatomical and

physiological measuremenls vary: also the measurements made by the

physicist,chemist,botanist, and agronomist. Statistics has been spoken of

as the study crfvariation. Fisher ( 1970) has observed,

The conccpljon of \lalr\tics !s lhe sludy of varialion i\ lhe naturnl oulcomc ofvie\ jnB lhe

subjecl as the sluJy of population\: for a population ol rndividuals in all re\pecis idenlical is

completely described by a descript()n of any one individual. logether wilh the number in r.he

group. The populationswhich are the object of slatistical sludy always display variation in

one or more rcspects.

The experimental scienlisl is frequentlyconcernedwith the different cir'

cumstances.conditions.or sources whichcontributeto the variation in the

measurements

he or she obtains. The analysis of variance(Chapter15)developedby Fisher is an important statistical procedurewhereby the variationin a set of experimental data can be partitionedinto componentswhichfrequentlymay be attributed to different c.lusal circumstances.

How may the variation in any set of measurenentsbe described?

THEMEAN OIPVANON


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Consider rhe lbllowing measL

Sample,4 I0

] 15 lt

SampleB I

8 15 2:

We note that the t\r,osanrpleInspectionindicates.however,variabie than those in sampAmong the possiblemeasuresthe mean deviation.and the rtbese is the slandarddeviation

5.2 THERANGEf$ .,9g"_ts_*.su:ptellJlg3l

metrts.ihe{angeis takej}-ar.th

[email protected]!0etrls. The rangefor

20 minus10,or 10. Therang

is 28 minusl, or 26. Themr

exhihit greater variation

thantla mucngreaterrange.Therar

samplesit is an unstabledesct

the rangefbr smallsamplesisdeviationbut increasesrapidlynot independentof samplesizedistributionsthattapert0 0 atttainingextremevaluesfor lar6rangescalculatedon samplescnotdirectlycomparable.Desoifectivelyusedin theapplication

5.3 THEMEANDEVIATIONConsiderthefbllowingmeasurer

Sample,4 It 8 8uSamplef I 4 7 IO ISamplca I 5

Intuitively,themeasurements

inwhichin rurn arelessvariablethrnI exhihitno variation atall.and 16. Ii we cxpressthemeasmeans.we obtain


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Considerthe lbllowing mea5uremerlts for two samplesr

Samplc.l t0 t: li llt t0SampleI I 8 t-5 tt t8

We note that the two salnples have the same mean, nantel), 15. Simple

inspection indicates. houevcr. that the measurements in sample B are morevariable than those in sample ,4: they difler more one liom another.Among the possiblemeasules used to describe this variation are the range,the mean deviation. and the standard deviation. The most imoortant oftheseis the standard dcviation.

5.2 IHE RANGE@arntio1, ln 43y1a!.qp,lc-Le

al_lrgalumentsthe rang is taken. aslhedifferencebetweenthe largesl andsmallestrl1g$]lrcmenl.s. The range for- lhe measurcments 10, ll, 5, I8, and lU rs

l0 minus 10.or 10. fhc range ft)r the mcasufemen(t l. l{, 15. ll, and l8

is ?8 minus I, or 26. The measurementsin th!' second set quite clearlyexhibit greatervariation than those in lhe first set, and thi5 reflects itself ina nuch gfa(er range. fhe rangehas two disadvantages. First, for largesamples it is an unstable descriptive measure. The rampling variance ofthe range fbr small samplcs is not much greater than that of the standarddeviatjon but increases rapidly with increase in N. Second,the range isnot independent of sample size. except under special circumstances. Fordistributionsthat taper to 0 at the extremities a bctter chance exists of obtainingextrenre values lbr large than lbr small samples. Consequently,

rangc; calculated on samplescomposed of diff'ercnt numbers of cases arenot directly comparable. Despitc thcse disadvantages the range may be ef'fectivclyused in the application of tests of significancc with small samples.

5.3 THEMEAN DEVIATIONConsider the following measuremcnts:

SamplcJ SlililiSSanplc B I I 7 l0 tlSample(' I5t0 15:e

lntuitively.the measurements in samplc,4are less variable than those in B.whichin turn are less variilblcthanthose in C. Indeed.the measurementsin z1 exhibit no variation at all. fhe means of the thre sarnplesare ll, 7.and i6. lf we cxpress thc measurementsas deviations fiom their samplemeans. we obtain


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61 MEASUPIS ANO(URTOSTS

OF VAnAION, srrWNtSS,

Sample,{ 0 0 000SampleI 3 0 +3 +6

-6-

SampleC 15 tt +1 +9 +13

Inspectionof these numbers suggeststhat as variation increases, thedepartureofthe observations from their sample mean increases. We may usethis characteristic to define a measure of variation. One such measureislhe mean deviation. The mean deviation is the arithmetic mean of the

absolute deviations from thearithmetic mean. An absolute deviation is adeviationwithout regard to algebraicsign. To obtain the mean deviationwe simply calculatethe deviations from the arithmetic mean, sumthese.disregardingalgebraicsign, and divide by N. For sample ,4 above, themean deviation is 0. For sample B the mean deviation is (6 + 3 +0+ 3 +6)/5: +:3.6. + 11 +4+

ForsampleCthemeandeviationis(159+13)/5:Y:10.4.The mean deviation is givenin algebraic language by the fbrmula

t5;rl ItD-:x,ltl

Here X -* is a deviation from the mean and l,f'- tl is a deviationwilhout regard to algebraic sign. The verticalbarsmeanthal signsareignored.

Hitherto, symbols above and below the summation sign ! have beenused to indicatethe limits of the summation. In the above formula for themean deviation these symbols have beenomitted, the summation beingclearly understood to extendover the N members in the sample. In thisand subsequent chapters symbols indicating the limits of summation will,for convenience, be omitted wherethese are understood clearly from thecontext to extend over N sample members. Where anypossibilityofdoubtcould exist, the symbolsaboveand belowthe summation sign will be inserted.

Themean deviation is infrequently used. It is not readily amenable toalgebraic manipulation. This circumstance stems from the useof absolutevalues. ln general,in statisticalwork lhe use of absolute values should beavoided,if at all possible.

The mean deviation is discussedhereprimarilyfor pedagogicreasons.It illDsfiatshow one parlicularmeasureof variation may be defined.

5,4 THESAMPLE ANO STANDARO VARIANCE DEVIAIION

Some of the deviations about the mean are posilive; others are negative.

The sum of deviations is 0. One method for dealing with the presenceof


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negative signs is to use the absolute deviations. as in tbe calculation of the

5.4 TfiESAMPITvAfl Nct At{D3meandeviationas ,ommendit. An althe deviationsabosquaresin thedefir

of the measuremermeanare-6, -3, (9, 0. 9. and36. Tl

The sumof sorriono]I?"tithe-yqri4lce-"tiifrBorh

!ry-dtviOlclrhe-surnumbero.fcases.I

t5.21

ln the illustrativee;deviationsaboutthe[5.2],iss': {ri: 13.

An altemativensquaresby N-I rivariancess, is given

t5.1i

Both formulas,Ivariance. Thesefc

processof plausibleItinctionis madehen

that definedby formr"

What is the essformula[5.2]andthe(heanswerto this qu

tinction was madebvalues,or parametenof a populationvariar>(X -X)" by N weshowa systematicte

divide> (X -t), byo:. Suchanestimatethanor lessthanqr.In all situations

required,the statisticN shouldbe used. Ibook an unbiasedesdescriptivestatistics,


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5.' IHT 5AMPIE VAFIANCE AND SIANDARD DEVIAIION

mean devialion as denncd in formula [5.1]. Thisprocedurehaslittle to recommendit. An altemative and generally preferable procedureis to squarethe deviations about the mean,sum these squares, and use this sum ofsquaresin the definition of a measure of variation. For example, the mean

olthe measurements l, 4,7, 10, and 13is 7. The deviations from the

-6, -3, 0,*3, and+6.

meanare The squareso[ these deviations are36,9, 0, 9, and 36. The sum of squares is 90.sum of squares of deviations about the mean is used in the defini

the variance-Both are in common use. blq .ttlS4_deli""" rhe variancebflivlOlqe-1h-stts-a{-sqlalq of deviatiops about thc mean bLN. thenq4b31-o!14.sqq.Denote this statistic by .r'':. Thus

-

2(.X x)',

ln the illustrative example in the paragraphabove, the sum of squares ofdeviationsaboutthe mean was 90. and the variance, according to formula[,5.2],isr':+!:18.

An altemative method of defining the varianceis to divide the sunl ofsquares by N -I rather than N. Thus. according to this dennition. thevariancess: is givenby

I5.31 -2

Both formulas, [5.2] and [5.3],providealternativedefinitionsof thevariance. These formulas have no derivation but are obtained by aprocessof plausiblereasoning. The readerwill note that no notational distinctjonismade here between the variancedefined by formula [5.2]andthatdefinedby formula [5.3].

What is the essential diference between the varianceas defined byformula[5.2]andthe varianceas defined by formulat5.31?To understandtheanswer to tbis questionthereader should recallthatin Chapter I a distinctionwasmade betu'een sample values, or estimates, and populationvalues,or parameters.Both formulas, [5.2] and [5.3],provideestimatesof a populationvalianceo!. For certain algebraic reasonswhen we dilide

-

:(.Y ,Y)' by N we obtaina biased estimate of rr". This estimate willshow a systematic tendencyto be less than 02. lt is biased. Whenwedivide!(-Y -t)' by N -l, however, we obtain an unbiased eslimate ofo:. Suchanestimatewill show no systematic tendency to be either greaterthan or less than or.

In all situations where an estimateof a populalionvariance rrr isrequired,theslatistic i'zwhich divides the sum of squares by N -I and not


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N should be used. In the greatmajority of situations discussed in thisbook an unbiased estimate is required. ln some situations involvingdescriptive statistics, convenienceand simplicity dictate the useof an es


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MIASURES Oa vARlAllON, SKEWNESS,

-

timatewhich divides the sum of squares hv N and not N l. lrt g


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differences X, Xr, X, ,1.,,. and X.3 X;. [n general, for Nmeasurmentsthe number of such dilferences is N(N- l)/2. To illustrale,fDr the measurements 1,4,7, 10, and ll, the differences between

-

eachmeasurementand every olher measurenent are -3, -6, -9, l:. ,1.

9, -3, -6, and -3.

6. Note that the sign of the difference depends onthe order of the measurements. If we obtain the sum of squares of the dit't'erencesbetween each meilsurement and every olher measurement anddivide by the number of such differences,the result is closely related to s';in fact it is simply twice r:. [n our example the sum of squares of dif[erencesis 450. We divide this by l0 to obtain 45.0. which is seen to betwice the variance. 22.5. as calculated by formula [-s.]1. In general,inalgebraicnoration it may- be shown that!.5 AN ItLUSIPAIVE APPUCAION

L5.4)

where the summatilferences.Thisresuleachvalueisfrom evdiferencesdividedb.

The varianceis afeet,then(X -x yr1.desirableto usea mei

units of the originalj

takingthesquarerootis called thestundard

t5.5i

or

t5.61

5.5 AN ITTUSIRATIVEAPI

Our understanding

ofwill beenrichedby cor

are of interest. Consieffectof a drug onacoof subjects, who receivthe drug, areused. Eiscoreson the codingts

Experimental S

7Conrrol 29


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36

The meanscorefor the-51.5. The investigatotmeansthatthedrughajects. The standarddeand 14.86, the experim

ancethan thecontrol grertrnga substantial inflrinfluenceon levelof pementaldatatheinvestigencesin the standardd

arithmeticmean.


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5.5 AN rrrUSlnATtVt 57APPCAION

-

:f'\'' -

l).qtX )'2\ '

N(N t)12

where the summation is understood to extend over N(N l)/l diflerences.This resuh meansthal s! is a descriptlve indexof how differenteachvalueis from every other value;in fact it is anaverageof the squareddifferencesdividedby 2.

-

The variance is a statistic in squared units. lf x X is a deviation infeet, then (X -X)' is a deviation in feet squared. For many purposesit isdesirableto use a measureof variationwhich is not in squared units butinunits of the original measurementsthemselves.We obtain this resuhbytakingthe square root of either formula [5.2]oformula[5.3]. This statistic

^fhus

is calledthestqndarddeyiation.

[5.-rI

5.5 AN ITTUSTRATIVE APPLICATION

Our understandingof the nature of the variance andthe standard deviationwill be enriched by considering illustrative situations wherethesestatisticsare of interest. Consider a simple experiment designed to investigate theeffecl of a dlug on a cognitive task suchas coding. An experimentalgroupof subjects,who receive the drug,and a control group,who do not receivelhe drug, are used. Each groupcontains l0 subjects. Let us assumethescoreson the coding task for the two groupsareas follows:

Experimental 5 7 11 ll 15 4'7 6E 85 96 99Control :9 .16 1'l 42 49 58 6t 63 69 7()

The mean score for the experimental groupis 50.0, and that for the control,

51.5. The investigator might be led to concludefrom inspecting thesemeansthat the drug had little or no effect on the performanceof the subjects.The standard deviations for the two groupsare, respectively. 35.63and 14.86, the experimentalgroupbeing much more variable in performancethanthe control group. Quiteclearly the treatment appearsto be exeninga substantial influenceon the variationin performance,althoughitsinfluenceon level of performanceis negligible. In the analysis ofexperimentaldata the investigator mustattendto, and if possibleinterpret. differencesin

the standard deviation, or variance,as well as differences in thearithmeticmean.


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SIANDAPOSCORES

68MaasuRls sKEwNEss, xuetosts

ot vARtAloN AND

the marks assigne

THE VARIANCEthe same as the

5.6 CATCUI.ATING SAMPTE AND THEFROM UNGROUPEDThis result follow['or purposesof calculation. it is convenient to write the variance and the

SIANDARDDEVIATION

DATA

correspondingobsstandard deviation in a different form. The variance may be $'ritten irs

meanof the origi

-A deviati,, >(,Y x)"X*c.

-

addedis then (X-

x t. Sinceth

-

>\xt + t! 2xr )

tion of a constant,

NI

trate,by adding a>.Y:+Nt,'_2N.t' we obtain6, 9, 127, and the mean o:r' -Nt'' 12. The deviation

-6, -3,0, +t, anr

If all measuretln this derivation note lhat the summation of X' over N is simply NX J: dard daviationis atl\o thc \umnl:Ltion ol 2XX i\ 2t:t' .-2Ntr. since >X: NX. The the standard devi

a

standarddeviation is given by


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tipliedby the conslis3x4:12. Toof a sample of meac is cX. A devia

squaring,summinl

Thus to calculate the standard deviation using this formula, we sum the

obtarn

squares of the original observations. subtract from this N times the squareof the arithmeticmean, divide by N- l. and then take the square root.For example,the five observations I, .1.7, 10. and l3 havea mean of 7.The squirres of these obscrvations are I, 6, 49, 100,and I 69. The sum ofthesesqrrared observations is 335. The variance is then

Thus if all measul

135 5 7r_

,.

.'._ ) Y-_rx' ---'"q11multipliedby c, an

'--.v

I -.-

is a negative numb

way of illustration

rrnd lhe:tantlaltl deviari,,rn i\ \ l=0 J.74.

varianceof 22.50,

An alternative formula for the standard deviation which avoids the

are multiplied by tl

calculation of the arithmetic mcan and may. ther!'[ore, be useful lbr certain

now 5 x 7, or 35.

compulational purposesi\

+30. Squaringtt/,\Tx,,-(:xr squaresis 2,250,

15.7l'

Y N(,V l) 23.72,whereas5The slight discrepaThisformularequiresoncoperationof division only.

ON THE DEVIAIION


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5.7 THEEFFECI STANDARD5.8 STANDARDSCOROFADDINGOR MUTTIPTYINGBYA CONSTANTHitherto we havec

I.l a

utultant i:t added to ull thc obsarr.'alionsin a surnple, tltc standurdthey wereoriginall.dcvitttion rttnuint uncltungcd. An examiner may conclude, for exarrple,X with meanX ar

that an cxamina(ion is too difficult. He may decide to add l0 points to all


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5.I SIANOARDSCORES 69themarks assigned. The standard deviation of the originalmarkswill bethe same as the standarddeviationof marks with the 10 pointsadded.This result follows directly from the fact that if X is an observation, thecorrespondingobservationwith the constant c addedis X + c. lf t is themean of the original observations, the mean with the constantaddedis

* + c. A deviation from the meanof the observations with the consranradded is then (,\'+ c) (t + c). which is readily observed to be equal toX -t. Sincethe deviations about the mean areunchangedby the additionof a constant,the standard deviationwill remainunchanged.To illustrate,by addinga constant, say, 5, to the measurements I, 4, 7, 10. and ll,weobtain6,9, 12, 15. and 18. The mean ofthe crriginalmeasurements

is

7. and the mean of the measurementswith ihe constantadded is 7 * 5. or12. The deviations from the mean are in both instances the same, namely,6,-3, 0, +3, and 1 6. The standard deviation in both instancesis 4.74.

lf all measuremants in a sample ore muhiplied by a (onstqnt,thestandardderiqtionisalsomultipliedby tht absolute value ofthat constont. lfthe standard deviation of examination marks is 4 and all marks are multipliedbythe constanl 3, then the standard deviation of the resulling marksis 3 x 4: 12. To demonstrate weobserve is the mean

this result. that iftof a sample of measurements, multiplied by

the mean of the measurementsr is cX. A deviationfrom the mean is Ihen rX -r'X , rX -.Yt. 81squaring.summingover N observations,and dividing by N- l. we

Obtain

st,Y-rY\2 c22(X -*)2

Thus if all measurenrents are muhiplied by a constant c, the variance ismulliplied by c': and the standarddeviation by the absolutevalue of c. If cis a negative number. say, 3, s is multiplied by theabsolute value 3. Byway of illustration,themeasurements

1,4,7, 10,13have a mean of7, avariance of 22.50, and a standard deviation of 4.74. If the measurementsare multiplied by theconstant5,we obtain 5. 20, l-5. -50, 65. The mean isnow-5x7,or 35. The deviationsfrom the mean are 30,-15,0,+15,*30. Squaringthesewe obtain 900, 225, 0, 225, 900. The sum ofsquares is 2,250, the varianceis 562.50, and the standarddeviation is23.72, whereas 5 times the original standard deviation of 4.74 is 23.70.The slightdiscrepancyresultsfrom the rounding of decimals.

5.8 STANDARDSCORESHithertowe haveconsideredscores or measurements in the form in which

theywereoriginallyobtained. Suchscores are represented by the symbol

X with meanX and standarddeviation s. Suchscores in their orisinal


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MEAsuREsoF vaprllroN, aNDKupTosrs

sxEwNEss,AOVANIAGESOF THE VARIANCTAN

form are spoken of as r.n! Jco/'eJ. We have also considered deviations In efect,

in relalion lo-

about the arithmetic mean. .r: X t. These are Inown as tleviatiott scoreof 65 ontheEngl.r( ore.r and have a mean of0 and a standard deviation of J. lf now we the mathemalics

examidivide the deviation about the mean by the standard deviation, we obtain tionabovethemean,thwhat is callcd a stundurd score represented by the symbol ;. Thus beconsideredto

be thethe mean,that is.52 +

XXx

individualmakesa scor

58on lhe mathematics

Standardscores have a mean of 0 and a standarddeviation of l. As pre

anceon the two subje(

viously shown, if ali measurementsin a sample are multiplied by a con-

his standardscoreis (5stant,the standard deviation is also multiplied by the absolutevalue of that

scoreis (58-52)lt2:

-

constant. Deviation scores. ,r: X X, have a standard deviation s. Each

darddeviationunitbelcscore has a constant -,\'added. This leavess unchanged. Ifall the devia

anceis .5 standard devtion scores are divided by ,r. which is the same thing as nrultiplying by the

individualdid muchmoconstanl l/.r, the standard deviation of the scores thus obtained is s/s: l.

lhe performanceof thTo illustrate. the following observations have been expressedin raw-

althoughthis is notrefle

score. deviation-score. and standard-score form,

orouscomparabilityof


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shouldbe identical in slIndividual ,' clearaswe proceed.The reader shouldn

/1 I 7l.llis equalto N -

R .61 l. Wec 1l-

D I.-._x(x

Et5 5 .'79 ,tIt) IO l.-58

The readershouldr

Sum .(J0 .{J0 sumof squaresof slandiMean l{J .0t) .ot)6.tl 6..11 l(xt

5.9 ADVANTAGESOI IHE \Becausestandardscores have zero mean and unit standarddeviation,

DEVIATIONAS MEASUI

they are readily amenable to certain forms of algebraic manipulation.Many formulationscan be derived more convenientlyusing standard

The variance and stand

scoresthan using raw or deviation scores.

measuresof variation.

The use of standardscoresmeans, in effect, that we are using the stan

variancehascerlainaddi

dard deviation as the unit of measurement. In the above exampleindivid

intoadditivecomponents

ualI is l.ll standard deviations, or standard deviation units, below the

cumstance.The sample

mean. while individual F is 1.58 standard devialionunitsabove the mean.

eslimateof thepopulatiot

Standardscores are frequently used to obtain compalability of obser

der certain assumptjons


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vations obtainedby different procedures. Consider examinations in

deviationin the populatiEnglishand mathematics applied to the same groupof individuals,and as-

doesof the meandeviati

sume the means and standard deviations to be as follows:deviationare more amemeasures.Theyenterinstatistics.Theyarewideon samplingstatisticsthe

Examination

Fnslish 658

effect the standarddevial

Malhemalics 5t l:

metersfrom samplevalr


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5.' ADVANIAGES O' IttE VARIANC! ANO SIANDAED DIVIAIION AS MEASURIS OF VATIA'ION71

ln cffect. in relation to the pcrlbrmance of the individuals in the group, ascore of 65 on the English cxamination is the equivalent of a score of -52 onthe mathematics examination. To illustrate. a score one stanclard deviation

above the mean.that is, 6-5'i 8, or 73. on the English examinutioncanhe consideled to be the equivalent of a score one standard deviation abovethe mean. that is.52* I2. or 64. on the mathematics cxamination. If anindividual makes a score of -57 on thc English examination and a score of58 on thc mathemalics examination, we may compare his relative performanceon rhc rwo subjects hv comparing his standard scores. On Englishhis standard score is (-57 65)/8 : 1.0.and on mathematics his standard

-

scor!'is (-5ll 5l)/l:: .-5. fhus on Englishhis pertbrmanccis one rtandarddeviation unit belo\\'the average, while on mathematics his perfornranceis

.-5 standard devialion unit abovc thc avcrage. Quite clearly.thisindividuai did much more poorly in English than in mathematicsrelativetoth perfbrmance of the group of irilividuals taking the examinations,althoughthis is not reflcctcd in the original marks assigned. To attain rigorouscomparabilityofscorcs,the distributions of scores on the two testsshould be identical in shape. The nreaning of this stat(-ment will hecomeclear as wc orocecd.

'fhe

reader shoulcl notc that thc sum of squarcs of standard scores, ):',is equal to N l. We obs!'r ve lhat .:' : (l ,l 1'/.r':hence

-

\-_, >(r tr :(.\' *t'

,,..,.^.'":*";T;*"', ;:ll:n;'i -r,; i )./N.,he

sum of squares of standard scorcs is N and not N l.

5.9 ADVANTAGESOF THE VARIANCEAND STANDARDDEVIAIIONASMEASURESOF VARIATION

The variance and standard deviation havc many advantages over othermeasures of variation. Much statistical work involves their use. Thevariancehas certain additive propertiesand may on occasion be partitionedinto additive components. each of rvhich may be related to some causal circumstance.The sample standard deviation is a more stable or accurateestimate of the population pantmeler than olher measures of virriation. Undercertain assumptions it provides a more stable estima(e of the standarddeviation in the population than the sample mean deviation. for example,does of tbe mean deviation in the population. The variance and standarddeviation are more amenable to mathematical manipulation than othermeasures. They enter into formulas for the computation of man) types ofstatistics. They are widely used as measures oferror. ln laterdiscussion

on sampling statistics the reader rvill observe that the stanrlar,l ellot is rneffect the stantlard deviation of errors made in estimating population parameters


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from sample values. These errors result from the operation of


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oF vARrATroN, AND

72MEAsuREs sKEwNEss,(uRTosrs

5.'I MEASURESOf SXEWNISSAND TUTT(

chance factors in random sampling. A full appreciation of the importance

The rationale for thiss

and meaning of the varianceand standard deviation in their many ramifica

tribution (or any set ofr

tions requires considelable familiarity with statistical ideas.

the mean, whenraisedbelowthe mean. whendistribution.4r.,: 0,an5,IOMOMENISAEOUTTHEMEAN sumsof deviations abr

power, will not balanceThe mean and the standard deviation are closely related to a family of g, + 0. lf the disrribtdescriptive statistics known as mom?nls. The first four moments about positive;whennegativrthe arithmetic mean are as follows: introducedin order to e

fer in variability. Thus

:{x -x)

Uskewnessof a set of n

15.8rDrt:Nscoreon a psychologica-

:(x t)2 N- ,will recallthata standar

'

"lt: N N uslngstandardscoresilmeasurementsto anotl_,,:rr,*rr" directlyanalogousto th

As an illustrationof

-2(X *)''n'

AI


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BI

ln general,the rth moment aboutthe mean is givenby

X)'Thesenumbersexpressr

-

t5.el^,._2(X

A

The term "moment" originatesin mechanics. Considera lever sup-Bportedby a fulcrum. lf a force li is applied to the lever at a distance-r.from the origin, then.l,r: is called the momentof the force. Further, ifa Set ,4is a symmetrical

deyiationsraisedtothetl

second force.ll is applied at a distance -rr. the total moment isfix1 *./l.rr.

lf we square th distances x, we obtain the second moment;if we cube

.4 64

them,we obtain the third moment; and so on. When we come to consider

a -64

frequency distributions, the origin is the analog of the fulcrum and the

frequenciesin the various class intervals are analogousto forces operating

For setl, ru,: 0 and g

at variousdistancesfrom the origin. Observe that the first moment about

.387. SetB is a positiv

the mean is 0 and the secondmoment is (N l)/N timestheunbiasedsam-

The commonly used

ple variance. The third momentis used to obtain a measureof skewness,

and is definedas

and the fourth moment.a measure of kurtosis.

t5.lrl

5,I1 MEASURES AND KURTOSISThis definitionis based

OF SKEWNESS

mean, when raisedto th(The commonly used measureof skewness makes use of the third moment fourthmoment.


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Thecoland is defined as tive thicknessof the tails

tionmaybeflatteror molI5.r0l meancontributemuchm

m2\ m2

The termzzr, is used to a


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5.11MIASURISOF S([WN[SS AND (URlOSrSThe rationale fbr this statistic is based on the observation that when a distribution(or any set ofnumbers) is symmetrical. the sum ofdeviations abovethe mean, u,hen raised to the third power. will balance the sum ofdeviationsbelow the mean. when raised to the third power. Thus for a symmetrical

distribution.,r r: 0. and the A,,: L If the distritrution is asymmetrical. thesums of deviations above and below the mean, when raised to the thirdpower. will not balance. Thus for an asymrnetrical distribution rrr,; 0 ande, + 0. lf the distribution.or set of numbers.is positivelyskewed.g, ispositive;when negatively skewed gr is negative. The quantit!,ar.f rri,isintroducedin order to ensure that gr is comparable for distributions that differin variability. Thus g, is independent ofthe scale ofmeasurement. Theskewnessof a set Df measurementsin gmms, meters.pounds,or units ofscore on a psychologicaltest can be directly compared usingg,. The reader

(,\ -tlir.

rvillrecallthata standard score is tlehned a\:: Oncreasonforusing standard scores is 1() achieve comparability of scores fiom one set ofmeasurementsto anothef. The use rrf ri:r r4 in the definirionof e, isdirectly analogous to the use ol's in the definition of a slandard score.

As an illustration of g,,considertq,o sets of numbers. ,4and I

A6 I0 l:t4R l0 l5

These numbers expressed as deviations tiom the mean become

Set,,1is a s1'mmetrical set of numbers. and set B is asymmetrical.

A410 rl +48420 +l -5-lhese

deviationsraised to the third power are as follows:

ItoB 64 o r l15

-8

For set I, rr':0 andg,:0. For set B.,rrr: 10.80,in.:9.10, and g,:.387. Set B is a positively skewed set of numbers.The commonly used measure of kurtosis involves the foufth moment.and is defined as

[s.l ]lThis definition is based on the observation that large deviations from themean. when raised to the fourth power. will contribute substantiallyto theti)urth moment. The concept of kurtosis is nrole closely Iinked to the relativethickness of the tails of distributions thiin to the idea that one distributionmay be flatter or more peakedthan another. Largedeviationsfrom the

mean contribute much more to thc fourth momenl than smallerdeviations.The term 2,,, is used to achieve comparability. lt serves the same purpose


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Sample standard deviiStandardscore.i

Momentsabout the mMeasureof skewness.Measureof kurtosis.

1

EXERCISES

LlFor the measuremmandeviation,(

i2 The variancecalcsum of squares ofu3

A biased varianceWhat is the corres

4The variance forvariancebe if all r

(b)divided by a c(-5 Show that )(X

Scholaslicaptitudr

of 100. A student

L.'

Expressthesescot

Expressthe measltr rn" sum of squa, /

l/8The mean andstarfor a class of 26 stmake scores of 50,scores?

9Calculatethesecol6, 10, 14, 16. Cor

l0 The following are

Group I 2 3Group II 2 4

Calculatemeasures


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ferguson - variability

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